Holographic Chern-Simons Theories

TL;DR

3D Chern–Simons theories provide a topological bulk whose boundary dynamics realize holography via edge states; conformal Chern–Simons gravity (CSG) and higher‑spin extensions realize AdS, Lobachevsky, and flat space holography through different boundary conditions and asymptotic algebras such as Virasoro, $\mathcal{W}$‑algebras, and BMS$_3$ with $c=-\bar{c}=12k$ or $c=24k$. The work highlights how various boundary conditions lead to distinct asymptotic symmetry algebras and unitarity constraints, including a flat‑space chiral gravity regime that connects to extremal CFTs like the Monster CFT at $k=1$. This framework provides a tractable laboratory for non‑AdS holography and higher‑spin holography, illustrating how holographic duals can arise beyond standard AdS/CFT and how edge‑mode dynamics encode bulk topological data. Overall, the results establish 3D CS formulations as versatile platforms for exploring holography across AdS, warped, and flat geometries, and for probing connections to extremal CFTs and higher spin dualities.

Abstract

Chern-Simons theories in three dimensions are topological field theories that may have a holographic interpretation for suitable chosen gauge groups and boundary conditions on the fields. Conformal Chern-Simons gravity is a topological model of 3-dimensional gravity that exhibits Weyl invariance and allows various holographic descriptions, including Anti-de Sitter, Lobachevsky and flat space holography. The same model also allows to address some aspects that arise in higher spin gravity in a considerably simplified setup, since both types of models have gauge symmetries other than diffeomorphisms. In these lectures we summarize briefly recent results.